Does the following infinite product converge to a non-zero value?


Mathworld gives a formula (ref 46 & 47) which, given that $1.127 > 1$, seems to imply that this is the case, but I may have misunderstood as I can't claim any knowledge of the Jacobi theta function used in (47).

  • $\begingroup$ Yes the given product converges to a positive value. The product you gave is related to Dedekind Eta function given by $$\eta(q) = q^{1/24}\prod_{n = 1}^{\infty}(1 - q^{n})$$ which converges absolutely if $|q| < 1$. Your product is equal to $(1.127)^{1/24}\eta(1/1.127)$. $\endgroup$ – Paramanand Singh Sep 9 '16 at 11:09
  • $\begingroup$ @Paramand Singh Yes, you are right, but I suspect the OP needs an answer at a more elementary level. $\endgroup$ – Vladimir Sep 9 '16 at 11:35

The series $\sum_{t=1}^\infty(1.127)^{-t}$ converges (the sum of a geometric progression with ratio $<1$), and hence, by the well-known theorem, your infinite product converges. (The definition of convergence of an infinite product includes the condition that the limit is nonzero.)


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